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BIFURCATION ANALYSIS OF A KALDOR-KALECKI MODEL OF BUSINESS CYCLE WITH TIME DELAY

Liancheng Wang 1 and Xiaoqin P. Wu 2

1 Department of Mathematics & Statistics Kennesaw State University

Kennesaw, GA 30144, USA e-mail: lwang5@kennesaw.edu

2 Department of Mathematics, Computer & Information Sciences Mississippi Valley State University

Itta Bena, MS 38941, USA e-mail: xpaul wu@yahoo.com

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday Abstract

In this paper, we investigate a Kaldor-Kalecki model of business cycle with delay in both the gross product and the capital stock. Stability analysis for the equilibrium point is carried out. We show that Hopf bifurcation occurs and pe- riodic solutions emerge as the delay crosses some critical values. By deriving the normal forms for the system, the direction of the Hopf bifurcation and the stabil- ity of the bifurcating periodic solutions are established. Examples are presented to confirm our results.

Key words and phrases: Kaldor-Kalecki model of business cycle, Hopf bifurcation, periodic solutions, stability.

AMS (MOS) subject classifications: 34K18

1 Introduction

In this paper, we study the Kaldor-Kalecki model of business cycle with delay of the following form:

( dY(t)

dt =α[I(Y(t), K(t))−S(Y(t), K(t))],

dK(t)

dt =I(Y(t−τ), K(t−τ))−qK(t), (1) whereY is the gross product,K is the capital stock, α >0 is the adjustment coefficient in the goods market, q ∈ (0,1) is the depreciation rate of capital stock, I(Y, K) and S(Y, K) are investment and saving functions, and τ ≥ 0 is a time lag representing delay for the investment due to the past investment decision.

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A business model in this line was first proposed by Kalecki [11], in which the idea of a delay of the implementation of a business decision was introduced. Later on, Kalecki [12] and Kaldor [10] proposed and studied business models using ordinary differential equations and nonlinear investment and saving functions. They showed that periodic solutions exist under the assumption of nonlinearity. Similar models were also analyzed by several authors and the existence of limit cycles were established due to the nonlinearity, see [4, 7, 23]. Krawiec and Szydlowski [14, 15, 16] combined the two basic models of Kaldor’s and Kalecki’s and proposed the following Kaldor-Kalecki model of business cycle:

( dY

(t)

dt =α[I(Y(t), K(t))−S(Y(t), K(t))],

dK(t)

dt =I(Y(t−τ), K(t))−qK(t).

This model has been studied intensively since its introduction, see [17, 19, 20, 21, 22, 24]. It is argued that a more reasonable model should include delays in both the gross product and capital stock, because the change in the capital stock is also caused by the past investment decisions [17]. Adding a delay to capital stock K leads to System (1).

As in [14], also see [1, 2, 22], using the following saving and investment functions S and I, respectively,

S(Y, K) = γY, I(Y, K) =I(Y)−βK

where β >0 andγ ∈(0,1) are constants, System (1) becomes the following system:

( dY(t)

dt =α[I(Y(t))−βK(t)−γY(t)],

dK(t)

dt =I(Y(t−τ))−βK(t−τ)−qK(t). (2) Kaddar and Talibi Alaoui [9] studied System (2). They gave a condition for the charac- teristic equation of the linearized system to have a pair of purely imaginary roots and showed that the Hopf bifurcation may occur as the delay τ passes some critical values.

However, they did not give the stability of the periodic solution and the direction of the Hopf bifurcation.

In this paper, we first give a more detailed discussion of the distribution of the eigenvalues of the linearized system of (2). So local stability of the equilibrium point is established. Conditions are found under which the Hopf bifurcation occurs and peri- odic solutions emerge as the delay crosses some critical values. By deriving the normal forms for System (2) using the normal form theory developed by Faria and Magalh˜aes [5, 6], the direction of the Hopf bifurcation and the stability of the bifurcating peri- odic solutions are established. Finally, some examples are presented to illustrate our theoretical results.

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2 Distribution of Eigenvalues

Throughout the rest of this paper, we assume that I(s) is a nonlinear function, C3, and that System (2) has an isolated equilibrium point (Y, K). Let I = I(Y), u1 = Y − Y, u2 = K −K, and i(s) = I(s +Y)−I. Then System (2) can be transformed into

( du1(t)

dt =α[i(u1(t))−βu2(t)−γu1(t)],

du2(t)

dt =i(u1(t−τ))−βu2(t−τ)−qu2(t). (3)

Let the Taylor expansion of iat 0 be

i(u) = ku+i(2)u2+i(3)u3+O(|u|4) where

k=i(0) =I(Y), i(2) = 1

2i′′(0) = 1

2I′′(Y), i(3) = 1

3!i′′′(0) = 1

3!I′′′(Y).

The linear part of System (3) at (0,0) is ( du1(t)

dt =α[(k−γ)u1(t)−βu2(t)],

du2(t)

dt =ku1(t−τ)−βu2(t−τ)−qu2(t), (4)

and its corresponding characteristic equation is

λ2+ [q−α(k−γ)]λ−αq(k−γ) + (βλ+αβγ)e−λτ = 0. (5) For τ = 0, Equation (5) becomes

λ2+ [q+β−α(k−γ)]λ−αq(k−γ) +αβγ = 0. (6) Define

k1 = βγ

q +γ, k2 = q+β α +γ,

and for the rest of the paper, we always assumek1 ≤k2.For the case thatk1 > k2, the discussion can be carried out similarly.

Theorem 2.1. Let τ = 0. Ifk < k1, all roots of Equation (6) have negative real parts, and hence(Y, K)is asymptotically stable. If k > k1, Equation (6) has a positive root and a negative root, and hence (Y, K) is unstable.

Now assume τ > 0. Let ωi (ω > 0) be a purely imaginary root of Equation (5).

After plugging it into Equation (5) and separating the real and imaginary parts, we have

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ω2+αq(k−γ) = αβγcos(ωτ) +βωsin(ωτ),

[q−α(k−γ)]ω = αβγsin(ωτ)−βωcos(ωτ). (7) Adding squares of two equations yields

ω4+ [q2−β22(k−γ)222q2(k−γ)2−α2β2γ2 = 0. (8) Let

A = q2−β22(k−γ)2, B = α2q2(k−γ)2−α2β2γ2.

If A≥0 and B ≥ 0, Equation (8) has no positive roots. If B <0,Equation (8) has a unique positive root

ω+ = s

−A+√

A2−4B

2 .

If A <0, B >0,and A2−4B >0, Equation (8) has two positive roots ω± =

s

−A±√

A2−4B

2 .

Solving Equation (7) for sin(ωτ) and cos(ωτ) yields

sin(ωτ) = ω3+ [αqk−α2γ(k−γ)]ω α2βγ2+βω2 , cos(ωτ) = α2qγ(k−γ) + (αk−q)ω2

α2βγ2+βω2 . Define

l1± = ω3±+ [αqk−α2γ(k−γ)]ω±

α2βγ2+βω±2 , l2± = α2qγ(k−γ) + (αk−q)ω2±

α2βγ2+βω±2 . We, thus, have the following result.

Lemma 2.1. Let ω± and l±i (i= 1,2) be defined above.

(i) If B < 0, then there exists a sequence of positive numbersj+}j=0 such that τ0+ < τ1+ < τ2+ < · · · < τj+ < · · ·, and Equation (5) has a pair of purely imaginary roots ±iω+ when τ =τj+.

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(ii) If A < 0, B > 0, and A2 −4B > 0, then there exist two sequences of positive numbersj+}j=0 andj}j=0 such that τ0+ < τ1+ < τ2+ <· · ·< τj+<· · · , τ0<

τ1 < τ2 < · · · < τj < · · · , and Equation (5) has a pair of purely imaginary roots ±iω± whenτ =τj±.

Here τj± (j = 0,1,2,· · ·) are defined below τj± = 1

ω±

arccosl±2 + 2jπ, if l±1 >0, 2π−arccosl±2 + 2jπ, if l±1 <0.

Define λ(τ) =σ(τ) +iω(τ) to be the root of Equation (5) such thatσ(τj±) = 0 and ω(τj±) =ω±, respectively.

Lemma 2.2. Let σ(τ) and τj± be defined above. Then σj+)>0, σj)<0.

Proof. Differentiate Equation (5) with respect to τ yields dλ

−1

= [2λ+q−α(k−γ)]eλτ +β λβ(λ+αγ) − τ

λ and a calculation gives

Re dλ

−1

τ=τj± = 2ω±22(k−γ)2+q2−β2

β22γ2±2) = 2ω±2 +A β22γ22±) which gives

Re dλ

−1

τ=τj± = ±√

A2−4B β22γ22±), completing the proof.

To discuss the distribution of the roots of Equation (5), we will need the following lemma due to Ruan and Wei [18].

Lemma 2.3. Consider the exponential polynomial P(λ, e−λτ) =p(λ) +q(λ)e−λτ

where p, q are real polynomials such that deg(q)<deg(p) and τ ≥ 0. As τ varies, the total number of zeros of P(λ, e−λτ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

Now we turn our attention to the relationship between A, B and our system pa- rameters. We look at the following two cases.

Case I. β≤q. In this case, A≥0.

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1. B ≥0⇐⇒k ≥k1 ork ≤ −βγ/q+γ; 2. B <0⇐⇒ |k−γ|< βγ/q.

Case II.β > q. In this case,

1. A ≥0, B ≥ 0⇐⇒k ≥max{p

β2−q2/α+γ, k1} or k ≤ min{−p

β2−q2/α+ γ,−βγ/q+γ};

2. B <0⇐⇒ |k−γ|< βγ/q;

3. A <0, B >0⇐⇒βγ/q <|k−γ|<p

β2−q2/α.

The discussions above, Theorem 2.1 and Lemmas 2.1, 2.2 and 2.3 imply the following Lemma 2.4.

Lemma 2.4. Assume β ≤q. Let τj+ be defined in Lemma 2.1. Then we have

(i) if B ≥ 0, then all roots of Equation (5) have negative real parts when k <

−βγ/q+γ and Equation (5) has roots with negative real parts and roots with positive real parts when k > k1;

(ii) if B <0, or |k−γ|< βγ/q, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ0+); Equation (5) has a pair of purely imaginary roots ±iω+ and all other roots have negative real parts whenτ =τ0+; it has2(j+1)roots with positive real parts and all other roots have negative real parts when τ ∈ (τj+, τj+1+ ), j = 0,1,2,· · · .

Lemma 2.5. Assume β > q. Let τj± be defined in Lemma 2.1. Then we have

(i) if A ≥ 0, B ≥ 0, then all roots of Equation (5) have negative real parts when k <min{−p

β2−q2/α+γ,−βγ/q+γ},and Equation (5) has roots with negative real parts and roots with positive real parts when k >max{p

β2−q2/α+γ, k1}; (ii) if B < 0, or if |k−γ|< βγ/q, all roots of Equation (5) have negative real parts for allτ ∈ [0, τ0+);Equation (5) has a pair of purely imaginary roots ±iω+ and all other roots have negative real parts whenτ =τ0+; it has2(j+1)roots with positive real parts and all other roots have negative real parts when τ ∈ (τj+, τj+1+ ), j = 0,1,2,· · · .

(iii) ifA <0, B >0andA2−4B >0,then we haveβγ/q <|k−γ|<p

β2−q2and A2−4B >0. Assume thatA2−4B >0. If −p

β2 −q2/α+γ < k <−βγ/q+γ, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ0+), Equation (5) has roots with positive real parts when τ ∈ (τ0+, τm) where m is the smallest positive integer such that τm > τ0+, it has a pair of purely imaginary roots±iω+

and all other roots have negative real parts when τ = τ0+. if τm < τ1+, Equation

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(5) has two roots with positive real parts and all other roots have negative real parts when τ ∈ (τ0+, τm) and all roots of Equation (5) have negative real parts when τ ∈ (τm, τ1+). If k1 < k < p

β2−q2/α+γ, Equation (5) has roots with negative real parts and roots with positive real parts.

The following Hopf bifurcation theorems follow immediately.

Theorem 2.2. Assume β ≤q. Let τj+ be defined in Lemma 2.1. Then we have

(i) the equilibrium point (Y, K) is asymptotically stable for all τ ≥ 0 when k <

−βγ/q+γ and it is unstable for all τ ≥0 when k > k1;

(ii) the equilibrium point (Y, K) is asymptotically stable for all τ ∈ [0, τ0+) and unstable for all τ > τ0+ when |k −γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y, K) whenτ =τj+ for j = 0,1,2,· · · .

Theorem 2.3. Assume β > q. Let τj± be defined in Lemma 2.1. Then we have

(i) the equilibrium point (Y, K) is asymptotically stable for all τ ≥ 0 when k <

min{−p

β2−q2/α + γ,−βγ/q + γ}, and unstable for all τ ≥ 0 when k >

max{p

β2−q2/α+γ, k1};

(ii) the equilibrium point (Y, K) is asymptotically stable for all τ ∈ [0, τ0+) and unstable for all τ > τ0+ when |k − γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y, K) whenτ =τj+ for j = 0,1,2,· · · .

(iii) Assume A2−4B >0. The equilibrium point(Y, K)is asymptotically stable for all τ ∈[0, τ0+) and unstable when τ ∈(τ0+, τm) where m is defined in Lemma 2.5 when−p

β2−q2/α+γ < k <−βγ/q+γ. System (2) undergoes a Hopf bifurca- tion at(Y, K)whenτ =τj+ forj = 0,1,2,· · · .Whenk1 < k <p

β2−q2/α+γ, the equilibrium point (Y, K) is unstable. System (2) undergoes a Hopf bifurca- tion at (Y, K) when τ =τj± for j = 0,1,2,· · · .

3 Direction and Stability of Hopf Bifurcation

From Section 2, we know that at (Y, K) the characteristic equation of linearized System (2) has a pair of purely imaginary roots±iω± if τ =τj± for each j under some conditions. Under these conditions, as the delay τ passes the critical values τj±, Hopf bifurcation occurs and periodic solutions emerge. In this section, by deriving a normal form for System (2) using a normal form theory developed by Faria and Magalh˜aes [5, 6], we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.

We first normalize the delay in System (2) by rescaling t→t/τ to get the following system

du1(t)

dt = ατ[(k−γ)u1(t)−βu2(t) +i(2)u21(t) +i(3)u31(t)] +O(|u1|4),

du2(t)

dt = τ[ku1(t−1)−βu2(t−1)−qu2(t) +i(2)u21(t−1) +i(3)u31(t−1)] +O(|u1|4).

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Let τcj± and τ = τc +µ. Then µ is the bifurcation parameter for System (9) and System (9) becomes





du1(t)

dt = α(τc+µ)[(k−γ)u1(t)−βτ u2(t) +i(2)u21(t) +i(3)u31(t)]

+O(|u1|4),

du2(t)

dt = (τc+µ)[ku1(t−1)−βu2(t−1)−qu2(t) +i(2)u21(t−1) +i(3)u31(t−1)] +O(|u1|4).

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The linearization of System (10) at (0,0) is ( du

1(t)

dt =ατc[(k−γ)u1(0)−βu2(0)],

du2(t)

dtc[ku1(−1)−βu2(−1)−qu2(0)]. (11) Let

η(θ) = Aδ(θ) +Bδ(θ+ 1) where

A=τc

α(k−γ) −αβ

0 −q

, B =τc

0 0 k −β

. Let C=C([−1,0],C2) and define a linear operator Lon C as follows:

Lϕ = Z 0

−1

dη(θ)ϕ(θ), ∀ϕ∈C.

Then System (10) can be transformed into

X(t) =˙ LXt+F(Xt, µ),

where X = (u1, u2)T, Xt=X(t+θ), θ∈[−1,0], and F(Xt, µ) = (F1, F2)T where F1 = α[(k−γ)µu1(0)−βµu2(0) +τci(2)u21(0) +τci(3)u31(0)] + h.o.t.,

F2 = kµu1(−1)−βµu2(−1)−qµu2(0) +τci(2)u31(−1) +τci(3)u21(−1) + h.o.t., where “h.o.t” represents high order terms. Write the Taylor expansion of F as

F(ϕ, µ) = 1

2F2(ϕ, µ) + 1

3!F3(ϕ, µ) + h.o.t..

Take the enlarged space of C

BC ={ϕ: [−1,0]→C2 : ϕ is continuous on [−1,0), ∃ lim

θ→0ϕ(θ)∈C2}. Then the elements of BC can be expressed asψ =ϕ+X0ν,ϕ ∈C and

X0(θ) =

0, −1≤θ <0, I, θ= 0,

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where I is the identity matrix on C and the norm of BC is |ϕ+X0ν| = |ϕ|+|ν|C2. Let C1 = C1([−1,0],C2). Then the infinitesimal generator A : C1 → BC associated with L is given by

Aϕ= ˙ϕ+X0[Lϕ−ϕ(0)] =˙

ϕ,˙ −1≤θ <0, Aϕ(0) +Bϕ(−1), θ = 0, and its adjoint

Aψ=

−ψ,˙ 0< s≤1,

ψ(0)A+ψ(1)B, s= 0, for ∀ψ ∈C1∗,

where C1∗ = C1([0,1],C2∗). Let C =C([0,1],C2∗) and forϕ ∈C and ψ ∈ C, define a bilinear inner product between C and C by

hψ, ϕi = ψ(0)ϕ(0)− Z 0

−1

Z θ 0

ψ(ξ−θ)dη(θ)ϕ(ξ)dξ

= ψ(0)ϕ(0) + Z 0

−1

ψ(ξ+ 1)Bϕ(ξ)dξ.

From Section 2, we know that ±iτcω0 are eigenvalues of A and A, where ω0 = ω+

or ω. Now we compute eigenvectors of A associated with iτcω0 and eigenvectors of A associated with −iτcω0. Let q(θ) = (ρ, k)Tecω0θ be an eigenvector ofA associated with iτcω0. Then Aq(θ) =iτcω0q(θ). It follows from the definition of A that

−α(k−γ)τc +iτcω0 αβτc

−kτce−iτcω0 βτce−iτcω0+qτc+iτcω0

q(0) = 0.

We can obviously choose q(θ) = (ρ, k)Tecω0θ where ρ=β+ (q+iω0)ecω0. Similarly, we can find an eigenvector p(s) of A associated with −iτcω0

p(s) = 1

D(σ, αβ)ecω0s, whereσ =−βecω0 −q+iω0

with D being a constant to be determined such that hp(s), q(θ)¯ i= 1. In fact, since hp(s), q(θ)¯ i= 1

D¯[kαβ(1 + (ρ−β)e−iτcω0) +ρ¯σ]

we have D = kαβ(1 + (¯ρ−β)ecω0) + ¯ρσ. Let P be spanned by q,q¯and P by p,p.¯ Then C can be decomposed as

C =P ⊕Q where Q={ϕ ∈C :hψ, ϕi= 0,∀ψ ∈P}.

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Let Q1 = Q∩C1. Let Φ(θ) = (q(θ),q(θ)) and Ψ(s) =¯ p(s)p(s)¯

. Then ˙Φ = ΦJ and Ψ =˙ −JΨ where J = diag(iτcω0,−iτcω0). Define the projectionπ :BC →P by

π(ϕ+X0ν) = Φ[(Ψ, ϕ) + Ψ(0)ν].

Let u= Φx+y, namely

u1(θ) = ecω0θρx1 +e−iτcω0θρx¯ 2+y1(θ), u2(θ) = ecω0θkx1+e−iτcω0θkx2+y2(θ).

Then System (10) can be decomposed as

x˙ =Jx+ Ψ(0)F(Φx+y, µ),

˙

y=AQ1y+ (I−π)X0F(Φx+y, µ).

This can be rewritten as

x˙ =Jx+12f21(x, y, µ) + 3!1f31(x, y, µ) + h.o.t.,

˙

y=AQ1y+ 12f22(x, y, µ) + 3!1f32(x, y, µ) + h.o.t., (12) where

fj1(x, y, µ) = Ψ(0)Fj(Φx+y, µ), fj2(x, y, µ) = (I−π)X0Fj(Φx+y, µ).

According to the normal form theory due to Faria and Magalh˜aes [5, 6, 8], on the center manifold, System (12) can be transformed as the following normal form:

˙

x=Jx+1

2g21(x,0, µ) + 1

3!g31(x,0, µ) + h.o.t.

wheregj1(x,0, µ) is a homogeneous polynomial of degreej in (x, µ). LetY be a normed space and j, p∈N. Let

Vjp(Y) =

 X

|q|=j

cqxq :q ∈Nq

0, cq ∈Y

 with norm|P

|q|=jcqxq|=P

|q|=j|cq|Y. DefineMj to be the operator inVj4(C2×kerπ) with the range in the same space by

Mj(p, h) = (Mj1p, Mj2h),

where (Mj1p)(x, µ) = [J, p(·, µ)](x) = Dxp(x, µ)Jx−Jp(x, µ). It is easy to check that Vj3(C2) = Im(Mj1)⊕Ker(Mj1) and

Ker(Mj1) = {µlxqek: (q,¯λ) =λk, k = 1,2, q∈N2

0,|(q, l)|=j}.

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Hence

ker(M21) = Span

µx1

0

, 0

µx2

, ker(M31) = Span

x21x2

0

, µ2x1

0

, 0

x1x22

, 0

µ2x2

. Define

31(x,0, µ) =f31(x,0, µ) + 3

2[(Dxf21)(x,0, µ)U21(x, µ) + (Dyf21)(x,0, µ)U22(x, µ)]

where

U21(x, µ)|µ=0 = (M21)−1ProjIm(M1

2)f21(x,0,0) = (M21)−1f21(x,0,0) and U22(x, µ) is determined by

(M22U22)(x, µ) =f22(x,0, µ).

Then

g21(x,0, µ) = Projker(M1

2)f21(x,0, µ), g13(x,0, µ) = Projker(M1

3)31(x,0, µ).

Let us compute g21(x,0, µ) first. Since 1

2f21(x,0, µ) = a1µx1+a2µx2+a20x21+a11x1x2+a02x22

¯

a2µx1+ ¯a1µx2+ ¯a02x21+ ¯a11x1x2+ ¯a20x22

! ,

where

a1 = −α

D¯[kβ(q+ (β−ρ)e−iτcω0) + ¯σ(k(β−ρ) +γρ)], a2 = −α

D¯[kβ(q+ ¯σ+βecω0)−ρ(kβe¯ cω0 + (k−γ)¯σ)], a20 = αρ2τc

D¯ i(2)(e−2iτcω0β+ ¯σ), (13) a11 = 2α|ρ|2τc

D¯ i(2)(β+ ¯σ), a02 = αρ¯2τc

D¯ i(2)(e2iτcω0β+ ¯σ), then

1

2g21(x,0, µ) = 1

2Projker(M1

2)f21(x,0, µ) =

a1µx1

¯ a1µx2

.

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Next we compute 3!1g31(x,0, µ) = 3!1Projker(M1

3)31(x,0, µ). Since the term O(µ2|x|) is irrelevant to determine the generic Hopf bifurcation, we have

1

3!g31(x,0, µ) = 1

3!Projker(M1

3)31(x,0, µ) = 1

3!ProjS31(x,0,0) +O(µ2|x|)

= 1

3!ProjSf31(x,0,0) + 1

4ProjS[(Dxf21)(x,0,0)U21(x,0) + (Dyf21)(x,0,0)U22(x,0)] +O(µ2|x|).

where

S= Span

x21x2

0

, 0

x1x22

. Step 1. Compute 3!1Projker(M31)f31(x,0,0). Since

1

3!f31(x,0,0) =

a30x31+a21x21x2 +a12x1x22+a03x32

¯

a03x31+ ¯a12x21x2 + ¯a21x1x22+ ¯a30x32

where

a30 = αρ3τc

D¯ [i(3)(βe−3iτcω0 + ¯σ)], a21 = 3α|ρ|2ρτc

D¯ [i(3)(βe−iτcω0 + ¯σ)], a12 = 3α|ρ|2ρτ¯ c

D¯ [i(3)(βecω0 + ¯σ)], a03 = αρ¯3τc

D¯ [i(3)(βe3iτcω0 + ¯σ)], we have

1

3!Projker(M1

3)f31(x,0,0) =

a21x21x2

¯ a21x1x22

.

Step 2. Compute 12ProjS[Dxf21(x,0,0)U21(x,0)]. The elements of the canonical basis of V22(C2) are

x21 0

,

x1x2

0

, x22

0

, µx1

0

, µx2

0

, µ2

0

, 0

x21

, 0

x1x2

,

0 x22

,

0 µx1

,

0 µx2

,

0 µ2

, whose images under 1

0M21 are, respectively x21

0

,− x1x2

0

,−3 x22

0

, 0

0

,−2 µx2

0

, µ2

0

, 3

0 x21

,

0 x1x2

,−

0 x22

,2

0 µx1

,

0 0

, 0

µ2

.

(13)

Hence

U21(x,0) = 1 iω0

a20x21−a11x1x213a02x22

1

302x21+ ¯a11x1x2 −¯a20x22

,

and 1

2ProjS[Dxf21(x,0,0)U21(x,0)] =

C1x21x2

1x1x22

where

C1 = 2 iω0

(2|a02|2−3a20a11+ 3|a11|2)

= −iα2|ρ|2ρτc2(i(2))2

12 ¯D|D|2 [6 ¯Dρ¯|β+σ|2−3Dρ(β+ ¯σ)(βe−2iτcω0 + ¯σ) + D¯¯ρ|βe−2iτcω0 +σ|2].

Step 3. Compute 12ProjS[(Dyf21)(x,0,0)U22(x,0)], where U22(x,0) is a second-order homogeneous polynomial in (µ, x1, x2) with coefficients in Q1. Let

h(x)(θ) =U22(x,0) =h20(θ)x21+h11(θ)x1x2+h02(θ)x22. The coefficients hjk = (h1jk, h2jk)T are determined by M22h(x) =f22(x,0,0) or

Dxh(x)Bx−AQ1(h(x)) = (I−π)X0F2(Φx,0) which is equivalent to

h(x)˙ −Dxh(x)Bx= ΦΨ(0)F2(Φx,0), h(x)(0)˙ −Lh(x) =F2(Φx,0),

where ˙h denotes the derivative of h(x)(θ) with respect to θ. Note that F2(Φx,0) =A20x21+A11x1x2 +A02x22

where

A20 = (2i(2)αρ2τc,2i(2)ρ2τce−2iτcω0)T, A11 = (4i(2)α|ρ|2τc,2i(2)α|ρ|2τc)T, A02 = (2i(2)αρ¯2τc,2i(2)ρ¯2τce2iτcω0)T.

Comparing the coefficients of x21, x1x2, x22 of these equations, it is not hard to verify that ¯h02 =h20,¯h11=h11 and that h20, h11 satisfy the following equations

20−2iτcω0h20= ΦΨ(0)A20,

20(0)−Lh20=A20, (14)

(14)

and

11 = ΦΨ(0)A11,

11(0)−Lh11 =A11. (15)

Noting that f21(x,0,0) = ΨF2(Φx,0), we deduce 1

2(Dyf21)h(x,0,0) =

ατci(2)

2 ¯D [β(ρe−iτcω0x1+ ¯ρecωox2)h1(−1) + ¯σ(ρx1 + ¯ρx2)h1(0)]

ατci(2)

2D [β(ρe−iτcω0x1+ ¯ρecωox2)h1(−1) + ¯σ(ρx1 + ¯ρx2)h1(0)]

where

h1(−1) = h120(−1)x21 +h111(−1)x1x2+h102(−1)x2, h1(0) =h120(0)x21+h111(0)x1x2+h102(0)x22. and hence

1

2ProjS[(Dyf21)h](x,0,0) =

C2x21x2

2x1x22

, where

C2 = ατci(2)

D¯ [e−iω0τ0βρh111(−1) +ρ¯σh111(0) +e0τ0βρh¯ 120(−1) + ¯ρ¯σh120(0)].

Here h20, h11 are determined by System (14) and System (15). After long but basic calculations, we obtain

h120(0) =

(2i(2)αρ2e−3τcω0( ¯D(−ie2iτcω0(2De3iτcω0σ)ω(−iq+ 2ω0) +kαβ(β+e2iτcω0σ)((−1 +e2iτcω0)β−2iecω0ω0)) +ecω0(β+e2iτcω0σ)(−iecω0kαβ+ie3iτcω0kαβ

+2βω0+ 2e2iτcω0(q+ 2iω00)¯ρ) +D(2ecω0ρ(β+e2iτcω0(q+ 2iω0))ω0

−ikαβ((−1 +e2iτcω0)β+ρ−2e2iτcω0ρ−2ie3iτcω0ω0))(β+ecω0σ))¯

/(ω0D(−iβ+e2iτcω0(−iq+ 2ω0))−α(βγ−ie2iτcω0(k−γ)(−iq+ 2ω0)) ¯D), h120(−1) =

2i(2)αρ2e−5τcω0

ω0|D|2 [i(−1 +e2iτcω0)(e2iτcω0(β+e2iτcω0σ) ¯Dρ¯+Dρ(β+e2iτcω0σ))¯ +( ¯D(−ie2iτcω0(2De2iτcω0ω0(−iq+ 2ω) +kαβ(β+e2iτcω0σ)((−1 +ecω0

−2ie2iτcω0ω0)) +ecω0(β+e2iτcω0)(−iecω0kαβ+ie3iτcω0kαβ+ 2βω0

+2e2iτcω0(q+ 2iω00)¯ρ) +D(2e2iτcω0ρ(β+e2iτcω0(q+ 2iω0))ω0

−ikαβ((−1 +e2iτcω0)β+ρ−e2iτcω0 −2ie3iτcω0))(β+e2iτcω0σ))]¯

/(2ω0(−iβ+e2iτcω0(−iq+ 2ω))−α(βγ−ie2iτcω0(k−γ)(−iq+ 2ω0)))

(15)

and

h111(0) =

4i(2)|ρ|2e−iτcω0

(βγ+ (−k+γ)q)|D|2[ecω0D(¯ −Dq+kαβ(β+σ)(−1 +ecω0βτc) +(β+σ)(−q+β(−1 +ecω0kατc))¯ρ)−D(ecω0(kαβ−(β+q)ρ) +kαβ(−β+ρ)τc)(β+ ¯σ)],

h111(−1) 4i(2)|ρ|2e−iτcω0

|D|2 [−e−2iτcω0ατc((β+σ) ¯Dρ¯+De−2iτcω0ρ(β+ ¯σ))

− 1

βγ+ (−k+γ)q( ¯Decω0(−Dq+kαβ(β+σ)(−1 +ecω0βτc)

−(β+σ)(−q+β(−1 +ecω0kατc))¯ρ)−D(ecω0(kαβ−(β+q)ρ) +kαβ(−β+ρ)τc)(β+ ¯σ))].

Collecting the results above, we obtain 1

3!g31(x,0, µ) =

b21x21x2

¯b21x1x22

+O(µ2|x|),

where b21 = a21+ 12(C1 +C2). Therefore, System (10) can be transformed into the following normal form:

1 =iτcω0x1+a1µx1 +b21x21x2+ h.o.t.,

˙

x2 =−iτcω0x2+ ¯a1µx2+ ¯b21x1x22+ h.o.t., (16) wherea1is given in (13). Letx1 =w1+iw2, x2 =w1−iw2andw1 =rcosξ, w2 =rsinξ.

Then (16) can be further written as

r˙=aµr+br3 + h.o.t., ξ˙=τcω0+ h.o.t.,

where a = Re[a1] and b = Re[b21]. Hence the first Lyapunov coefficient is l1(µ) = b+O(µ), see [3, 13].

Theorem 3.1. Let a and b be given above.

(i) The bifurcating periodic solution is stable if b <0, and unstable if b >0;

(ii) The Hopf bifurcation is supercritical if ab <0, and subcritical if ab > 0.

(16)

Remark. The coefficient a is given by a= Re[a1] =

k2α2(q202)

β22γ220)|D|2[−αω022−q2−ω02)(β2γ+ (k−γ)(q202)) +α3(−β4γ32γ(k2−3kγ+ 2γ2)(q202) + (k−γ)3(q220)2042q−(q202)(q−ω02))−α4(k−γ)2((k−γ)2q(1 +qτc)(q202)

−β2γ2(q+q2τccω20)) +α2ω02(2kγ(−β2q+ 2q3+q4τc+ 2qω02−τcω40) +k22q−2q3−q4τc−2qω2cω04) +γ2(−(q202)(2q+q2τc−τcω02) +β2(2q+q2τccω02))))].

Although the explicit algorithm is derived to compute b, it is difficult to determine the sign of b for general α, β, γ, k, q. But if i(2) = 0, it is easy to see C1 = C2 = 0 and hence b can be simply expressed as

b = −3i(3)

|D|22+q202+ 2βqcos(τcω0)−2βωsin(τcω0))(−kαβ2q+ 2β2q +2β2q2+q4−kαβ2q2τc+ 2β2ω20+ 2q2ω2−kα2β2τcω2004

+β(β2q+ 3q(q202)−kα(q2+q3τc−ω02+qτcω20)) cos(τcω0) +β2(q2−ω20) cos(2τcω0)−β3ω0sin(τcω0) + 2kαβqsin(τcω0)

−3βq2ω0sin(τcω0)

+kαβq2τcω0sin(τcω0)−3βω03sin(τcω0)−2β30sin(2τcω0)).

4 Numerical Simulations

In this section, we give some examples to illustrate the theoretical results obtained in the previous sections.

Example 1. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and I(s) = tanh(0.5s).

Then (0,0) is an equilibrium point of System (2),k = 0.5, i(2) = 0, i(3) =−0.041667.

Hence ω+ = 0.6066, and τ0+ = 3.1382. Take τ = 2.5. According to Theorem 2.2 (ii), the trivial equilibrium point (0,0) is asymptotically stable, (Figure 1).

Example 2. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and I(s) = tanh(0.5s).

(17)

0 10 20 30 40 50 60 70 80 90 100

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

t

Y(t)

0 10 20 30 40 50 60 70 80 90 100

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

t

K(t)

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Y

K

Figure 1: The equilibrium point (0,0) is asymptotically stable when τ < τ0+.

0 10 20 30 40 50 60 70 80 90 100

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

t

Y(t)

0 10 20 30 40 50 60 70 80 90 100

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

t

K(t)

−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Y

K

Figure 2: The stable periodic orbit generated by Hopf bifurcation when β < q.

0 10 20 30 40 50 60

−0.02

−0.015

−0.01

−0.005 0 0.005 0.01 0.015

t

Y(t)

0 10 20 30 40 50 60

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15

t

K(t)

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

−0.1

−0.05 0 0.05 0.1

Y

K

Figure 3: The stable periodic orbit generated by Hopf bifurcation when β > q.

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